The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many students... The class has thirty students. Twenty students study physics while twelve study both physics and chemistry. If eight students study neither physics nor chemistry, how many students study chemistry?
Understand the Problem
The question is a word problem that involves calculating the number of students studying chemistry in a class, given information about the number of students studying physics, both subjects, and neither subject. We can use the principle of inclusion-exclusion to solve it.
Answer
14 students study chemistry.
Answer for screen readers
14 students study chemistry.
Steps to Solve
- Find the number of students studying either physics or chemistry
The total number of students is 30, and 8 study neither subject. Therefore, the number of students studying either physics or chemistry is: $30 - 8 = 22$
- Apply the Principle of Inclusion-Exclusion
The principle of inclusion-exclusion states that:
$|P \cup C| = |P| + |C| - |P \cap C|$
Where: $|P \cup C|$ is the number of students studying either physics or chemistry (or both). $|P|$ is the number of students studying physics. $|C|$ is the number of students studying chemistry. $|P \cap C|$ is the number of students studying both physics and chemistry.
- Plug in the values and solve for $|C|$
We know that $|P \cup C| = 22$, $|P| = 20$, and $|P \cap C| = 12$. Substitute these values into the equation:
$22 = 20 + |C| - 12$
- Isolate $|C|$
Simplify the equation:
$22 = 8 + |C|$
Subtract 8 from both sides:
$|C| = 22 - 8$ $|C| = 14$
14 students study chemistry.
More Information
The principle of inclusion-exclusion is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets.
Tips
A common mistake is forgetting to account for the students studying both subjects, which leads to overcounting those students. Another mistake is not subtracting the students who study neither subject from the total number of students, leading to an incorrect calculation of the number of students studying at least one of the subjects.
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